# Topological susceptibility in QCD and corresponding pole

The topological susceptibility in QCD (here I've used path integral approach, and hence I will neglect all contact terms) is defined as $$\kappa (p) \equiv \lim_{y \to 0}\int d^{4}x e^{ip(x-y)}Q(x)Q(y)e^{iS} \equiv \lim_{y \to 0}\int d^{4}xe^{ip(x-y)}\langle 0| Q(x)Q(y)|0\rangle ,$$ where $$Q(x) \equiv \frac{1}{32 \pi}G \tilde{G} \equiv \partial_{\mu}^{x}K^{\mu}(x)$$ is the topological charge of QCD.

It can be shown, after some manipulations, that $$\tag 1 \kappa (0) = -c\langle 0|\bar{u}u|0\rangle ,$$ where $c$ is constant (which is zero if at least one of quarks is massless).

From the other hand, from the definition we have that $$\tag 2 \kappa_{0} \equiv \lim_{p \to 0}p_{\alpha}p_{\beta}\Pi^{\alpha \beta}(p),$$ where $$\Pi^{\alpha \beta}(p) \equiv \int d^{4}x e^{ipx} \langle 0| K^{\alpha}(x)K^{\beta}(0)|0\rangle$$ By combining $(1)$ and $(2)$ we have the statement that $$\lim_{p \to 0}p_{\alpha}p_{\beta}\Pi^{\alpha \beta}(p) = -c\langle 0| \bar{u}u|0\rangle \neq 0$$ So we conclude that $P^{\alpha \beta}(p)$ has the pole. However, this doesn't fixed uniquely the pole structure of $P^{\alpha \beta}$: in general, $$\lim_{p \to 0}\Pi_{\alpha \beta}(p) = -a\frac{g_{\alpha \beta}}{p^{2}} + b\frac{p_{\alpha}p_{\beta}}{p^{4}}, \quad a + b = c\langle 0| \bar{u}u|0\rangle$$ This ambiquity is important, since if $a \neq 0$, then, for example, the gluon propagator obtains positive valued correction in denominator, $$\tag 3 \lim_{p \to 0} D_{\mu \nu}(p) \sim \frac{g_{\mu \nu}}{p^{2} + \frac{a\kappa (0)}{p^{2}}},$$ which describes confinement phenomena qualitatively, while if $a = 0$, then there is no correction.

So how to establish the value of $a$ in $(3)$?

• Ia the physical meaning of topological correlator $\Pi_{\alpha\beta}$ something to do with the instanton tunneling event? [Because the $Q$ is related to the instanton number, and $K^\alpha$ looks like its current?] – annie heart May 29 '18 at 1:55

1) Note that $p^\alpha p^\beta \Pi_{\alpha\beta}$ is gauge invariant, but $\Pi_{\alpha\beta}$ is not. This implies that $a$ and $b$ are not separately gauge invariant, only their sum (or difference, in your convention) is. For example, Diakonov and Eides [Sov. Phys. JETP 54 (2), 232-240] write down a spectral representation corresponding to $a=0$, but many other authors implicitly assume $b=0$.

2) Also note that there is no model independent relation between the gluon propagator and the topological correlator $\Pi_{\alpha\beta}$.

• Ia the physical meaning of topological correlator $\Pi_{\alpha\beta}$ something to do with the instanton tunneling event? [Because the $Q$ is related to the instanton number, and $K^\alpha$ looks like its current?] – annie heart May 29 '18 at 1:54